We begin our discussion of potentials by considering the two homogeneous equations. For example, if we wish to associate a
potential with
such that
is the result of differentiating
the potential, we observe that we can satisfy GLM by construction
if we suppose a vector potential such that:

(8.28)

In that case:

(8.29)

as an identity.

Now consider FL. If we substitute in our expression for
:

(8.30)

We can see that if we define:

(8.31)

then

(8.32)

is also an identity. This leads to:

(8.33)

Our next chore is to transform the inhomogeneous MEs into
equations of motion for these potentials - motion because MEs (and
indeed the potentials themselves) are now potentially dynamical
equations and not just static. We do this by substituting in the
equation for
into GLE, and the equation for
into AL. We
will work (for the moment) in free space and hence will use the
vacuum values for the permittivity and permeability.

The first (GLE) yields:

(8.34)

The second (AL) is a bit more work. We start by writing it in terms of
instead of
by multiplying out the :